Question

which statement best describes $f(x) = -2\sqrt{x - 7} + 1$?\
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• $-6$ is in the domain of $f(x)$ and in the range of $f(x)$.\
• $-6$ is not in the domain of $f(x)$ but is in the range of $f(x)$.\
• $-6$ is neither in the domain of $f(x)$ nor in the range of $f(x)$.\
• $-6$ is in the domain of $f(x)$ but not in the range of $f(x)$.

Explanation:

Step1: Check Domain of \( f(x) = -2\sqrt{x - 7}+1 \)

The expression under the square root must be non - negative. So, \( x - 7\geq0 \), which implies \( x\geq7 \). Since \( - 6<7 \), \( -6 \) is not in the domain of \( f(x) \).

Step2: Check Range of \( f(x) \)

The square root function \( \sqrt{x - 7}\geq0 \). Multiply by - 2: \( - 2\sqrt{x - 7}\leq0 \). Then add 1: \( -2\sqrt{x - 7}+1\leq1 \). Let's see if \( y=-6 \) can be in the range. Set \( -2\sqrt{x - 7}+1=-6 \). Subtract 1 from both sides: \( -2\sqrt{x - 7}=-7 \). Divide both sides by - 2: \( \sqrt{x - 7}=\frac{7}{2} \). Square both sides: \( x - 7=\frac{49}{4} \), so \( x=\frac{49}{4}+7=\frac{49 + 28}{4}=\frac{77}{4}=19.25\geq7 \). So when \( x = \frac{77}{4} \), \( f(x)=-6 \), which means \( -6 \) is in the range of \( f(x) \).

Answer:

-6 is not in the domain of \( f(x) \) but is in the range of \( f(x) \)